Some Envelope Theorems for Integer and Discrete Choice Variables

Though the envelope theorem is a widely used tool of applied economic analysis, the standard version of the theorem can only be used if all of the choice variables are assumed to be continuous. This limitation is significant because the natural description of many economic choice variables is as integers (e.g., the number of projects or children, as well as variables that represent various yes-no choices and choices that entail fixed costs). A continuous representation of such variables is not only unsatisfactory and a source of potential error, but it can also make certain kinds of economic analyses intractable or unproductive. This paper shows that modified but intuitive versions of the envelope theorem can be used with integer or discrete choice variables, provided the optimization problem satisfies the usual conditions. Thus, the results presented here make it possible to use the envelope theorem in a variety of economic problems